Time-multiplexed dual atomic magnetometry

ABSTRACT

Time-multiplexed atomic magnetometry uses first and second atomic vapor cells located adjacent to a sample to be measured. Each vapor cell operates according to a sequence of alternating pumping and probing stages. However, the sequences are temporally offset from each other such that the second vapor cell is pumped while the first vapor cell is probed, and the first vapor cell is pumped while the second vapor cell is probed. With this time-multiplexed operation, the magnetic field generated by the sample can be measured without any time gaps. The Hilbert transform of the signals may be taken to obtain their instantaneous phases, which may then be interleaved to form a single gapless time sequence that represents the magnetic field of the sample over a time window that lasts for several continuous pumping/probing stages.

RELATED APPLICATIONS

This application claims priority to U.S. Provisional Patent ApplicationNo. 63/057,815, filed Jul. 28, 2020 and titled “Time-Multiplexed DualMagnetometry”, the entirety of which is incorporated herein byreference.

BACKGROUND

In atom magnetometry, spin-polarized atoms in an external magnetic fieldprecess at the Larmor frequency. This precession is typically measuredvia Faraday rotation, i.e., the spin-polarized atoms rotate the linearpolarization of a weak laser beam that passes through the atoms. Thisrotation of the laser polarization can be detected with a polarimeter,and the resulting electronic signal can be processed into a value of theexternal magnetic field. Since the gyromagnetic ratio of the atoms(i.e., the Larmor frequency per unit magnetic-field strength) isdetermined primarily by the energy-level structure of the atoms, atomicmagnetometry benefits from high accuracy, as compared to other forms ofmagnetometry (e.g., SQUID, fluxgate, Hall effect, magnetoresistance,etc.).

In addition, recent advances in the development of millimeter-size vaporcells has enabled the placement of atom-vapor-based magnetometer headsclose to the sample to be measured (e.g., within one centimeter of thesample). This improves sensitivity since the magnetic field strengthgenerated by a sample drops at least as 1/r, where r is the distancebetween the sample and the magnetometer head. In some cases (e.g., amagnetic dipole), the magnetic field drops as rapidly as 1/r³, furtheremphasizing the need to miniaturize sensor heads so that they can beplaced closer to the sample.

SUMMARY

An atomic magnetometer is typically operated in an alternating sequenceof pumping and probing stages. In each pumping stage, a pump laser beamis directed through a vapor cell to spin-polarize gaseous atoms therein.An external magnetic field may be applied to establish a quantizationaxis. The polarization (e.g., circular or linear), propagationdirection, and modulation (e.g., AM or FM) of the pump laser beam isselected such that the optical pumping results in a ground-statecoherence (i.e., a coherence between two or more magnetic ground-statesublevels of the atoms). During the probing stage, a linear polarizedprobe laser beam passes through the vapor cell. The spin-polarized atomsrotate the polarization of the probe laser beam synchronously with theLarmor precession. The rotated polarization is then measured with apolarimeter.

The duration of each probing stage is limited by a dephasing time of theatoms (i.e., the transverse spin relaxation time T₂). Specifically,collisions between atoms disrupt the ground-state coherences, washingout the Larmor precession, and hence the measured signal. Althoughtechniques exist for increasing the dephasing time (e.g.,spin-relaxation coatings and buffer gasses), dephasing times are stilltypically on the order of milliseconds. Furthermore, as the size of thevapor cell decreases, the rate of collisions between the atoms and thewalls increases, leading to shorter dephasing times (even if the wallshave a spin-relaxation coating).

The duration of each pumping stage is typically between a few hundredmicroseconds and a few milliseconds, depending on the available power ofthe pump laser beam and the transition strengths of the atomic speciesused. Thus, for each cycle of one pumping stage followed by one probingstage, the atoms may be measured for as little as 50% of the time. Thatis, half of the time is wasted preparing the atoms, which limits thesignal-to-noise ratio (SNR). Gaps in the measured time record of theatoms can also introduce aliasing and other deleterioussignal-processing artifacts that mask the true magnetic signal to bemeasured.

To solve these problems, the present embodiments feature systems andmethods for time-multiplexed atomic magnetometry performed with twovapor cells located adjacent to (e.g., on opposite sides of) the sampleto be measured. The first vapor cell is operated according to a firstsequence of alternating pumping and probing stages. Similarly, thesecond vapor cell is operated according to a second sequence ofalternating pumping and probing stages. However, the second sequence isdelayed relative to the first sequence such that the second vapor cellis pumped while the first vapor cell is probed, and the first vapor cellis pumped while the second vapor cell is probed. With thistime-multiplexed operation, the magnetic field generated by the samplecan be measured without any time gaps. More specifically, and asdescribed in more detail below, the signals from the two vapor cells canbe interleaved to form a single gapless time sequence that representsthe time-varying magnetic field generated by the sample over the entiretime sequence.

By using two vapor cells, the present embodiments advantageously havetwice the signal-to-noise ratio of conventional atomic magnetometersthat use only one vapor cell (assuming equal vapor pressures, vapor cellsizes, atomic species, etc.). However, the present embodiments alsooffer advantages over simply doubling the size or pressure of one vaporcell. For example, doubling the vapor cell size results in the extraatoms being located farther from the sample, where they are lesssensitive to the magnetic field. As a result, doubling the vapor cellsize does not necessarily double the SNR. On the other hand, in thepresent embodiments the two vapor cells may be located on opposite sidesof the sample, in which case both vapor cells are located proximate tothe sample, ensuring equal sensitivity to the magnetic field. Increasingthe vapor pressure inside the cell can help, although the resultingpressure broadening can reduce T₂. More than two vapor cells can be usedto achieve even greater increases in signal-to-noise ratio.

In embodiments, a time-multiplexed dual atomic magnetometer operates asa pair of free-induction-decay atomic magnetometers. In theseembodiments, the signal from each of the two vapor cells is continuouslyrecorded over several oscillations. For a single probing stage, theresulting signal is approximately equal to an exponentially decayingsinusoid, which can be fitted to extract a center frequency which equalsthe average Larmor frequency over the probing phase. The Larmorfrequency may then be converted into a corresponding value of themagnetic field. Repeating this process over several consecutive cyclesproduces a time sequence of magnetic field values. The bandwidth of thisapproach is limited by the duration of one cycle. However, the sequencecan be used to identify low-frequency components spanning over severalcycles.

To increase the bandwidth, instantaneous-phase retrieval may beimplemented on the recorded signals. This technique was recentlydemonstrated for atomic magnetometers in “Wide-bandwidth atomicmagnetometry via instantaneous-phase retrieval” by N. Wilson et al.(arXiv:2003.04526v1), although it has been used in geosciences forseveral decades. For example, see “The calculation of instantaneousfrequency and instantaneous bandwidth” by A. E. Barnes (Geophysics 57,1520-1539, 1992). In addition to these references, details aboutinstantaneous phase and frequency can be found in “Estimating andinterpreting the instantaneous frequency of a signal. I. Fundamentals”by B. Boashash (Proc. IEEE 80, 520-538, 1992). With instantaneous-phaseretrieval, the present embodiments are expected to operate at bandwidthsexceeding 10 kHz.

The present embodiments may be used to enhance magnetometry in a host ofapplications, including geosciences, magnetic communication, threatdetection, the measurement of bio-magnetic signals (e.g.,magnetoencephalography), nuclear magnetic resonance (NMR), and magneticresonance imaging (MRI). In another application, the present embodimentsare used to measure the time-varying magnetic field generated bymolecules in an aqueous solution. It is hypothesized that the motion ofthese molecules (e.g., stretching, rotating, translating, etc.) causeselectric charges therein to accelerate, which produces magnetic fieldsat the femtotesla level. When the time-varying magnetic field is later“played” to cells (e.g., by applying electric currents to coils toreplicate the time-varying magnetic field), the cells may behave as ifthe original molecules were present. In this regard, the time-varyingmagnetic field may be used to replicate the pharmacological effects of acompound on the cells, but without physically exposing the cells to theactual compound. As such, the present embodiments may be used toidentify new therapies for treating cancer. Examples of molecules whosetime-varying magnetic fields can be measured and subsequently used forsuch therapeutic purposes include small interfering RNA (siRNA) andmessenger RNA (mRNA) from genes.

In embodiments, a time-multiplexed dual atomic magnetometer includesfirst and second vapor cells located adjacent to a sample cell such thata magnetic field generated by a sample within the sample cell inducesLarmor precession of atoms within the first and second vapor cells. Thedual atomic magnetometer also includes a first polarimeter that measuresa first polarization of a first probe beam after the first probe beampropagates through the first vapor cell. The first polarimeter outputs afirst polarization signal indicative of the first polarization. The dualatomic magnetometer also includes a second polarimeter that measures asecond polarization of a second probe beam after the second probe beampropagates through the second vapor cell. The second polarimeter outputsa second polarization signal indicative of the second polarization. Thetime-multiplexed dual atomic magnetometer also includes a signalprocessor that processes alternating data blocks of the first and secondpolarization signals to generate a single gapless temporal sequence thatrepresents the magnetic field generated by the sample.

In other embodiments, a method for time-multiplexed dual atomicmagnetometry includes generating, by a sample within a sample cell, amagnetic field that induces Larmor precession of atoms within first andsecond vapor cells located adjacent to the sample cell. The method alsoincludes measuring, with a first polarimeter, a first polarization of afirst probe beam after the first probe beam propagates through the firstvapor cell. The first polarimeter outputs a first polarization signalindicative of the first polarization. The method also includesmeasuring, with a second polarimeter, a second polarization of a secondprobe beam after the second probe beam propagates through the secondvapor cell. The second polarimeter outputs a second polarization signalindicative of the second polarization. The method also includesprocessing alternating data blocks of the first and second polarizationsignals to generate a single gapless temporal sequence that representsthe magnetic field generated by the sample.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a top sectional view of a dual atomic magnetometer thatmeasures magnetic fields generate by a sample, in an embodiment.

FIG. 2 is a side section view of the dual atomic magnetometer of FIG. 1.

FIG. 3 shows first and sequence timing sequences that illustratetime-multiplexed operation of the dual atomic magnetometer of FIGS. 1and 2, in an embodiment.

FIG. 4 shows how instantaneous-phase retrieval is implemented with eachdata block of FIG. 3 to obtain a corresponding instantaneous-phaseblock, in an embodiment.

FIG. 5 shows how several consecutive frequency blocks may beconcatenated into the single gapless temporal sequence, in anembodiment.

FIG. 6 is a side section view showing the dual atomic magnetometer ofFIGS. 1 and 2 being used with a bias field oriented perpendicularly tothe propagation direction of probe beams, in an embodiment.

FIG. 7 is a side sectional view of components of the dual atomicmagnetometer of FIGS. 1 and 2 in which a charge moves along the +zdirection within a bias field oriented in the +y direction.

FIG. 8 illustrates how an alternating sequence of the data blocks can beprocessed to generate a single gapless temporal sequence for thescenario depicted in FIG. 7, in an embodiment.

DETAILED DESCRIPTION OF THE EMBODIMENTS

FIGS. 1 and 2 are top and side sectional views of a dual atomicmagnetometer 100 that measures magnetic fields generated by a sample110. The dual atomic magnetometer 100 includes first and second vaporcells 104(1), 104(2) filled with first and second atomic vapors 106(1),106(2), respectively. The sample 110 is placed inside a sample cell 102that is located between the first and second vapor cells 104(1), 104(2)in the x direction (see the right-handed coordinate system 120). A firstpump beam (e.g., see first pump beam 640(1) in FIG. 6) spin-polarizesthe first atomic vapor 106(1) by optically pumping the atoms of thefirst atomic vapor 106(1) into one or more ground-state magneticsublevels such that the atoms precess at a Larmor frequency. Thespin-precessing atoms, in turn, rotate the polarization of a linearlypolarized first probe beam 130(1) that passes through the first vaporcell 104(1). A first polarimeter 140(1) measures the polarization of thefirst probe beam 130(1) after exiting the first vapor cell 104(1),outputting a first polarization signal 142(1).

Similarly, a second pump beam (e.g., see second pump beam 640(2) in FIG.6) spin-polarizes the second atomic vapor 106(2) by optically pumpingthe atoms of the second atomic vapor 106(2) into one or moreground-state magnetic sublevels such that the atoms also precess at aLarmor frequency. These spin-polarized atoms rotate the polarization ofa linearly polarized second probe beam 130(2) that passes through thesecond vapor cell 104(2). A second polarimeter 140(2) measures thepolarization of the second probe beam 130(2) after exiting the secondvapor cell 104(2), outputting a second polarization signal 142(2).

The polarization of the first probe beam 130(1) oscillates at aninstantaneous Larmor frequency f_(L)(t), assuming that magnetic fieldgradients are negligible (i.e., the atoms in the first atomic vapor106(1) interacting with the first probe beam 130(1) are subjected to thesame magnetic field). The instantaneous Larmor frequency f_(L)(t)depends on the scalar magnitude of the magnetic field, which has twocomponents: a time-varying signal field {right arrow over(B)}^((s))(t)=(B_(x) ^((s))(t), B_(y) ^((s))(t), B_(z) ^((s))(t))arising from the sample 110, and a constant (i.e., time-independent)bias field {right arrow over (B)}⁽⁰⁾=(B_(x) ⁽⁰⁾, B_(y) ⁽⁰⁾, B_(z) ⁽⁰⁾).Thus, the instantaneous Larmor frequency f_(L)(t) can be representedmathematically as f_(L)(t)=γ|{right arrow over (B)}^((s))(t)+{rightarrow over (B)}⁽⁰⁾|/(2π), where γ is the gyromagnetic ratio of thespecies of the first atomic vapor 106(1). Assuming |{right arrow over(B)}^((s))(t)|<<|{right arrow over (B)}⁽⁰⁾|, the instantaneous Larmorfrequency f_(L)(t) is approximated by f_(L)(t)≈γ|{right arrow over(B)}⁽⁰⁾|/(2π), which is time-independent. Therefore, the bias field{right arrow over (B)}⁽⁰⁾ sets a nominal Larmor frequency f_(L)⁽⁰⁾≈γ|{right arrow over (B)}⁽⁰⁾|/(2π) that is subsequently modified bythe signal field {right arrow over (B)}^((s))(t). In FIGS. 1 and 2, thebias field {right arrow over (B)}⁽⁰⁾ is oriented along the +z direction(i.e., {right arrow over (B)}⁽⁰⁾=(0, 0, B_(z) ⁽⁰⁾), for which f_(L)⁽⁰⁾=γB_(z) ⁽⁰⁾/(2π). However, the bias field {right arrow over (B)}⁽⁰⁾may point in other directions, as discussed in more detail below. Thesame argument holds for the second probe beam 130(2) and the secondatomic vapor 106(2). It is assumed herein that the bias field {rightarrow over (B)}⁽⁰⁾ is the same at both of the vapor cells 104(1) and104(2) and the sample cell 102.

FIG. 1 also shows a signal processor 144 that implementstime-multiplexed operation of the dual atomic magnetometer 100. Thesignal processor 144 is a circuit that acquires and processes thepolarization signals 142(1) and 142(2) into a magnetic-field sequence{B_(j)}. Although not shown in FIG. 1, the signal processor 144 mayinclude a computing device with a processor and a memory storingmachine-readable instructions that, when executed by the processor,control the signal processor 144 to implement the functionalitydescribed herein. Alternatively, the signal processor 144 may be a chipor circuit (e.g., a field-programmable gate array) that has beenpreviously programmed to implement the functionality described herein.When the polarization signals 142(1) and 142(2) are analog electronicsignals, the signal processor 144 may include analog-to-digitalconverters that convert the polarization signals 142(1) and 142(2) intodigital electronic signals that are subsequently processed.Alternatively, each of the polarimeters 140(1) and 140(2) may include ananalog-to-digital converter, wherein the polarization signals 142(1) and142(2) are received by the signal processor 144 as digital electronicsignals. A reference oscillator 148 establishes common timing for dataacquisition, time tagging, and laser-timing control. The signalprocessor 144 may output the magnetic-field sequence {B_(j)} to datastorage (e.g., a memory card or hard drive), a computer monitor orscreen for display to a user, or another computer system (e.g., viaEthernet or Wi-Fi) for additional signal processing and storage.

In some embodiments, the signal processor 144 also serves as acontroller that outputs one or more timing signals 146 that control whenthe first and second pump beams and the first and second probe beams130(1), 130(2) pass through the vapor cells 104(1) and 104(2). Forexample, the timing signals 146 may be used to gate (i.e., turn on andoff) each of the pump beams and probe beams 130(1), 130(2) by driving acorresponding acousto-optic modulator, electro-optic modulator, ormechanical shutter. The timing signals 146 may also be used to changethe frequency of one or more of the pump beams and the probe beams130(1), 130(2). In other embodiments, a controller separate from thesignal processor 144 implements timing control of the first and secondpump beams and the first and second probe beams 130(1), 130(2).

FIG. 3 shows first and second timing sequences 300(1), 300(2) thatillustrate time-multiplexed operation of the dual atomic magnetometer100 of FIGS. 1 and 2. The first timing sequence 300(1) corresponds tooperation of the first vapor cell 104(1), the first probe beam 130(1),the first pump beam, and the first polarimeter 140(1). Similarly, thesecond timing sequence 300(2) corresponds to operation of the secondvapor cell 104(2), the second probe beam 130(2), the second pump beam,and the second polarimeter 140(2).

The first timing sequence 300(1) is formed from a first repeating frame302(1) that has: (i) a first pumping stage 304(1) with a first pumpingduration T_(p) ⁽¹⁾, (ii) a first measurement stage 306(1) with a firstmeasurement duration T_(m) ⁽¹⁾, and (iii) a first dead stage 308(1) witha first dead-time duration T_(d) ⁽¹⁾. During the first pumping stage304(1), the first probe beam 130(1) is blocked while the first pump beamspin-polarizes the first atomic vapor 106(1). During the firstmeasurement stage 306(1), the first pump beam is blocked while the firstprobe beam 130(1) propagates through the first atomic vapor 106(1). Thefirst polarimeter 140(1) measures the polarization of the first probebeam 130(1) to obtain a first data block 340(1) of the firstpolarization signal 142(1). During the first dead stage 308(1), no firstpolarization signal 142(1) is obtained (e.g., both the first probe beam130(1) and the first pump beam are blocked, or the output of the firstpolarimeter 140(1) is ignored). The first timing sequence 300(1) istherefore periodic with a first period T₁=T_(p) ⁽¹⁾+T_(m) ⁽¹⁾+T_(d) ⁽¹⁾,and has a measurement duty cycle η₁=T_(m) ⁽¹⁾/T₁.

The second timing sequence 300(2) is similar to the first timingsequence 300(1) except that it is delayed with respect to the firsttiming sequence 300(1) by a second dead-time duration T_(d) ⁽²⁾ of asecond dead stage 308(2). Specifically, the second timing sequence300(2) is formed from a second repeating frame 302(2) that has: (i) asecond pumping stage 304(2) with a second pumping duration T_(p) ², (ii)a second measurement stage 306(2) with a second measurement durationT_(m) ⁽²⁾, and (iii) the second dead stage 308(2). During the secondmeasurement stage 306(2), the first polarimeter 140(2) measures thepolarization of the second probe beam 130(2) to obtain a second datablock 340(2) of the second polarization signal 142(2). The second timingsequence 300(2) therefore is periodic with a second period T₂=T_(p)⁽²⁾+T_(m) ⁽²⁾+T_(d) ⁽²⁾, and has a measurement duty cycle η₂=T_(m)⁽²⁾/T₂.

The duration T_(d) ⁽²⁾ is selected such that the second pumping stage304(2) ends when the first measurement stage 306(1) ends. This allowsthe second measurement stage 306(2) to begin immediately when the firstmeasurement stage 306(1) ends, eliminating any gap between the datablocks 340(1) and 340(2). Similarly, the duration T_(d) ⁽¹⁾ is selectedsuch that the first pumping stage 304(1) ends when the secondmeasurement stage 306(2) ends. This allows the first measurement stage306(1) to resume immediately when the second measurement stage 306(2)ends, eliminating any gap between the second data block 340(2) and asubsequent third data block 340(3).

The polarization signal 142 within each data block 340 approximates anexponentially-decaying sine wave at the instantaneous Larmor frequency.The time constant of the exponential decay is determined by transversespin relaxation of the atoms in the vapors 106. The vapor cells 104 maybe filled with a buffer gas (e.g., N₂ or ⁴He) and/or lined with ananti-relaxation coating (e.g., paraffin) to reduce spin relaxation andincrease the time constant. Dephasing times T₂ are typically between afraction of a millisecond and several tens of milliseconds, depending onthe geometry and size of the vapor cells 104, the pressures of thevapors 106 and buffer gas (when included), the choice of atomic speciesfor the vapors 106 (e.g., Rb, Cs, K, Na, etc.), the choice of speciesfor the buffer gas (when included), the type of anti-relaxation coating(when included), etc. The dephasing time T₂ is the primary determinantof the measurement durations T_(m) ⁽¹⁾ and T_(m) ⁽²⁾, as thesignal-to-noise ratio decays with T₂.

If general, the first and second measurement durations T_(m) ⁽¹⁾, T_(m)⁽²⁾ do not need to be equal. Similarly, the first and second pumpingdurations T_(p) ⁽¹⁾, T_(p) ⁽²⁾ do not need to be equal. In someembodiments, the first and second measurement durations T_(m) ⁽¹⁾, T_(m)⁽²⁾ are equal, as shown in FIG. 3. In these embodiments, T_(d) ⁽¹⁾,T_(d) ⁽²⁾, T_(m) ⁽¹⁾, and T_(m) ⁽²⁾ are all similar. In someembodiments, the first and second pumping durations T_(p) ⁽¹⁾, T_(p) ⁽²⁾are similar.

FIGS. 4 and 5 illustrate how an alternating sequence of the data blocks340 can be processed to generate a single gapless temporal sequence 502that represents the signal field {right arrow over (B)}^((s))(t). InFIG. 4, instantaneous phase retrieval is implemented with each datablock 340(i) to obtain a corresponding instantaneous-phase block 440(i).Details about instantaneous-phase retrieval can be found in“Wide-bandwidth atomic magnetometry via instantaneous-phase retrieval”by N. Wilson et al. (arXiv:2003.04526v1), which is incorporated hereinby reference in its entirety. Specifically, the instantaneous phaseϕ_(l) ^((i))(t) is obtained mathematically as the argument of ananalytic phase ϕ_(a) ^((i))(t):

ϕ_(l) ^((i))(t)=arg(ϕ_(a) ^((i))(t))=arg(φ^((i))(t)+i

{φ ^((i))(t)}),  (1)

where φ^((i))(t) is the measured polarization angle of the data block340(i), and

{ } indicates a Hilbert transform. In FIGS. 4 and 5, each phase block440(i) shows ϕ_(l) ^((i))(t) after unwrapping. The derivative of theinstantaneous phase ϕ_(l) ^((i))(t), after unwrapping, gives theinstantaneous Larmor frequency for the data block 340(i):

$\begin{matrix}{{{f_{L}^{(i)}(t)} = \frac{1}{2\pi}}{\frac{d\;{\phi_{I}^{(i)}(t)}}{dt}.}} & (1)\end{matrix}$

The magnetic field sensed by the atoms in the vapor 106 during the datablock 340(i) is directly proportional to the instantaneous Larmorfrequency f_(L) ^((i))(t), as described previously.

In some embodiments, and as shown in FIG. 4, the unwound instantaneousphase ϕ_(l) ^((i))(t) of each instantaneous-phase block 440(i) is fit toa straight line (e.g., via linear regression) to obtain a correspondingslope m_(i) that represents the average value of dϕ_(l) ^((i))(t)/dt(i.e., the average Larmor frequency) over the measurement duration T_(m)^((i)) of the corresponding data block 340(i). Dividing mi by thegyromagnetic ratio γ gives a single corresponding magnetic-field valueB_(i). A sequence of several consecutive magnetic-field values {B_(i)}can then be used to identify changes in the magnetic field between datablocks 340(i). For example, the Fourier transform of the sequence{B_(i)} can be calculated to identify components of {right arrow over(B)}^((s))(t) with frequencies less than the measurement duration T_(m).

In some embodiments, and as shown in FIG. 5, the time-derivative of eachphase block 440(i) is calculated to obtain a corresponding frequencyblock 540(i) that numerically represents the instantaneous Larmorfrequency f_(L) ^((i))(t) of the phase block 440(i). For example, wheneach phase block 440(i) is represented as a temporal phase sequence of Ninstantaneous-phase values Φ^((i))={ϕ₁, ϕ₂, . . . , ϕ_(N)} equallyspaced in time by a point spacing Δt, then the corresponding frequencyblock 540(i) can be represented as a temporal frequency sequence of N−2values F^((i))={f_(j)=(ϕ_(j+1)−ϕ_(j−1)/(2Δt)} for j=2 to N−1. Othermethods of numerical differentiation may be used to calculate thetemporal frequency sequence from the temporal phase sequence (e.g., themethod of finite difference coefficients) without departing from thescope hereof. Such methods may also be used to obtain frequencies pointscorresponding to ϕ₁ and ϕ_(N) such that the frequency sequence F^((i))and the phase sequence Φ^((i)) have the same number of points, and thepoints are aligned in time.

As shown in FIG. 5, several consecutive frequency blocks 540(i) may beconcatenated into the single gapless temporal sequence 502.Equivalently, each frequency block 540 may be sequentially appended tothe temporal sequence 502 to extend the temporal sequence 502 in time.Each point of the temporal sequence 502 may then be divided by thegyromagnetic ratio γ to obtain a magnetic-field sequence {B_(j)} thatapproximates the time-varying total magnetic-field strength |{rightarrow over (B)}^((s))(t)+{right arrow over (B)}⁽⁰⁾|. The magnetic-fieldsequence {B_(j)} may be subsequently analyzed (e.g., Fourier transform)to identify features associated with the sample 110.

In other embodiments, the instantaneous-phase blocks 440(i) areconcatenated together to form the single gapless temporal sequence 502.The time derivative of the temporal sequence 502 may then be calculated,after which each point is divided by the gyromagnetic ratio γ to obtainthe magnetic-field sequence {B_(j)}. In these embodiments, concatenatingbefore the time derivative may improve estimates of the instantaneousfrequency at the boundaries of the phase blocks 440(i).

In the example of FIGS. 1 and 2, the dual atomic magnetometer 100 mayalso include a solenoid 114 to generate the bias field {right arrow over(B)}⁽⁰⁾=(0, 0, B_(z) ⁽⁰⁾) along the propagation direction of the probebeams 130(1), 130(2). The vapor cells 104(1), 104(2) and sample 110 arelocated along an axis of the solenoid 114 where the homogeneity of thebias field {right arrow over (B)}⁽⁰⁾ is greatest. The bias field {rightarrow over (B)}⁽⁰⁾=(0, 0, B_(z) ⁽⁰⁾) may be alternatively generated withone or other magnetic coils, such as a pair of Helmholtz coils.Furthermore, one or more layers of magnetic shielding 112 may surroundthe solenoid 114 (or other magnetic coils), the sample cell 102, and thevapor cells 104 to block external magnetic fields.

The instantaneous Larmor precession frequency f_(l)(t) is givenmathematically by

$\begin{matrix}\begin{matrix}{{f_{L}(t)} = {\frac{1}{2\pi}{{{{\overset{\rightarrow}{B}}^{(s)}(t)} + {\overset{\rightarrow}{B}}^{(0)}}}}} \\{= {\frac{\gamma}{2\pi}{\sqrt{\left( {B_{x}^{(0)} - {B_{x}^{(s)}(t)}} \right)^{2} + \left( {B_{y}^{(0)} - {B_{y}^{(s)}(t)}} \right)^{2} + \left( {B_{z}^{(0)} - {B_{z}^{(s)}(t)}} \right)^{2}}.}}}\end{matrix} & (3)\end{matrix}$

For B_(x) ⁽⁰⁾=B_(y) ⁽⁰⁾=0, Eqn. 3 simplifies to

$\begin{matrix}{{{f_{L}(t)} = {{\frac{\gamma}{2\pi}B_{z}^{(0)}\sqrt{1 + \frac{{{{\overset{\rightarrow}{B}}^{(s)}(t)}}^{2}}{\left( B_{z}^{(0)} \right)^{2}} - {2\frac{B_{z}^{(s)}(t)}{B_{z}^{(0)}}}}} \approx {\frac{\gamma}{2\pi}{B_{z}^{(0)}\left( {1 - \frac{B_{z}^{(s)}(t)}{B_{z}^{(0)}} + \ldots} \right)}}}},} & (4)\end{matrix}$

where the Taylor expansion in Eqn. 4 assumes |{right arrow over(B)}^((s))(t)|<<B_(z) ⁽⁰⁾. Eqn. 4 shows that f_(L)(t) approximatelyequals the nominal Larmor frequency f_(L) ⁽⁰⁾≈γB_(z) ⁽⁰⁾/(2π), but ismodulated primarily (i.e., to first order) by the z-component B_(z)^((s))(t) of the signal field {right arrow over (B)}^((s))(t).Equivalently, B_(x) ^((s))(t) and B_(y) ^((s))(t) only modulate theinstantaneous Larmor frequency f_(L)(t) to second order in the Taylorexpansion, and are therefore suppressed relative to B_(z) ^((s))(t).Accordingly, the setup shown in FIGS. 1 and 2 preferentially measuresthe z component B_(z) ^((s))(t) of the signal field {right arrow over(B)}^((s))(t).

FIG. 6 is a side section view of the dual atomic magnetometer 100 ofFIGS. 1 and 2 being used with a bias field {right arrow over (B)}⁽⁰⁾oriented perpendicularly to the propagation direction of the probe beams130(1) and 130(2). Specifically, the bias field {right arrow over(B)}⁽⁰⁾ points in the +y direction, i.e., {right arrow over (B)}⁽⁰⁾=(0,B_(y) ⁽⁰⁾, 0). The instantaneous Larmor precession frequency f_(l)(t) isgiven mathematically by

$\begin{matrix}{{f_{L}(t)} = {{\frac{\gamma}{2\pi}B_{y}^{(0)}\sqrt{1 + \frac{{{{\overset{\rightarrow}{B}}^{(s)}(t)}}^{2}}{\left( B_{y}^{(0)} \right)^{2}} - {2\frac{B_{y}^{(s)}(t)}{B_{y}^{(0)}}}}} \approx {\frac{\gamma}{2\pi}{{B_{y}^{(0)}\left( {1 - \frac{B_{y}^{(s)}(t)}{B_{y}^{(0)}} + \ldots} \right)}.}}}} & (5)\end{matrix}$

Now, B_(x) ^((s))(t) and B_(z) ^((s))(t) only modulate the instantaneousLarmor frequency f_(L)(t) to second order in the Taylor expansion, andare therefore suppressed relative to B_(y) ^((s))(t). Accordingly, thesetup shown in FIG. 6 preferentially measures the y component B_(y)^((s))(t) of the signal field {right arrow over (B)}^((s))(t). Similarcalculations show that when the bias field {right arrow over (B)}⁽⁰⁾points in the x direction, B_(y) ^((s))(t) and B_(z) ^((s))(t) onlymodulate the instantaneous Larmor frequency f_(L)(t) to second order,and are therefore suppressed. Thus, the dual atomic magnetometer 100 canbe used to preferentially measure a component of the signal field {rightarrow over (B)}^((s))(t) by aligning the bias field {right arrow over(B)}⁽⁰⁾ along the direction of the component.

FIG. 7 is a side sectional view of components of the dual atomicmagnetometer 100 of FIGS. 1 and 2 in which a charge 702 moves along the+z direction within a bias field {right arrow over (B)}⁽⁰⁾ oriented inthe +y direction. For clarity in FIG. 7, the sample cell 102 and vaporcells 104(1) and 104(2) are not shown. Due to its motion, the charge 702generates a magnetic field 730 that circles in the x-y plane. At thefirst probe beam 130(1), the magnetic field 730 adds to the bias field{right arrow over (B)}⁽⁰⁾, causing the atoms probed by the first probebeam 130(1) to precess at a Larmor frequency greater than γ|{right arrowover (B)}⁽⁰⁾|. At the at the second probe beam 130(2), the magneticfield 730 subtracts from the bias field {right arrow over (B)}⁽⁰⁾,causing the atoms probed by the second probe beam 130(2) to precess at aLarmor frequency less than γ|{right arrow over (B)}⁽⁰⁾|. Thus, due tothe magnetic field 730, atoms probed by the probe beams 130(1) and130(2) precess at different rates. In this case, the data blocks 340cannot be concatenated into the single gapless temporal sequence 502since the Larmor frequency shifts with each data block 340.

FIG. 8 illustrates how the alternating sequence of the data blocks 340can be processed to generate a single gapless temporal sequence 802 forthe scenario depicted in FIG. 7. Here, each data block 340(i) isprocessed into a corresponding instantaneous-phase block 440(i), asdescribed above (not shown in FIG. 8). Each phase block 440(i) is thenfit to a straight line to obtain a corresponding best-fit slope m_(i),also as described before and shown in FIG. 8. The best-fit slope m_(i)is then used to obtain a corresponding residual block 840(i) of thephase block 440(i), which approximates the signal field {right arrowover (B)}^((s))(t) without the static bias field {right arrow over(B)}⁽⁰⁾. For the data blocks 340 obtained from the second vapor cell104(2) (e.g., data blocks 340(l), where l is even), the correspondingresidual blocks 840 are inverted to account for the different directionsof the magnetic field 730 relative to the bias field {right arrow over(B)}⁽⁰⁾. Here, “inverted” means that the value of each point in theresidual block 840 is multiplied by −1, as represented in FIG. 8 by acircle with “−1” inscribed therein. The inverted residual blocks 840 arethen be interleaved with the uninverted residual blocks 840 obtainedfrom the first vapor cell 104(1) (e.g., data blocks 340(l), where l isodd) to form the single gapless temporal sequence 802. Thetime-derivative of the temporal sequence 802 is then calculated toobtain a corresponding frequency sequence, which is then divided by γ toobtain the magnetic-field sequence {B_(j)}.

Optical pumping of the vapors 106(1) and 106(2) may be implemented usinga technique known in the art. For example, when the bias field {rightarrow over (B)}⁽⁰⁾ is parallel to the propagation direction of the probebeams 130(1) and 130(2) (e.g., the z direction in FIGS. 1 and 2),linearly polarized first and second pump beams may be directed throughthe respective vapor cells 104(1) and 104(2) parallel to the probe beams130(1) and 130(2) to pump the atoms into a dark superposition ofground-state magnetic sublevels (i.e., coherent population trapping).The resulting ground-state coherence between these magnetic sublevelsvaries in time, and is equivalent to a precession of the atoms in thereference frame of the probe beams 130(1) and 130(2). Alternatively, thepump beams may be oriented perpendicularly to the probe beams 130(1) and130(2). For example, in FIG. 6 circularly polarized pump beams 640(1)and 640(2) may be directed through the respective vapor cells 104(1) and104(2) parallel to the bias field {right arrow over (B)}⁽⁰⁾ to opticallypump the atoms into a stretched state that processes along the zdirection in the reference frame of the probe beams 130(1) and 130(2).Another optical pumping technique known in the art may be used withoutdeparting from the scope hereof.

In some embodiments, more than two vapor cells 104 are placed around thesample cell 102. For example, in the dual atomic magnetometer 100 shownin FIG. 2, third and fourth vapor cells 104 may be placed above andbelow the sample cell 102 (in the y direction). In these embodiments,four polarization signals can be processed and interleaved, as describedabove, to form the single gapless temporal sequence 802 and themagnetic-field sequence {B_(j)}. Furthermore, while FIGS. 1 and 2 showthe vapor cells 104(1) and 104(2) located on opposite sides of thesample cell 102 (along the x direction), the vapor cells 104(1) and104(2) may be positioned otherwise without departing from the scopehereof. For example, the vapor cells 104(1) and 104(2) may be positionedproximate to adjacent perpendicular side faces of the sample cell 102.

Some embodiments include only the signal processor 144, wherein allother components (e.g., the vapor cells 104(1) and 104(2), the samplecell 102, the polarimeters 140(1) and 140(2), etc.) are provided by athird party. Other embodiments exclude the signal processor, 144, whichis provided by a third party.

Changes may be made in the above methods and systems without departingfrom the scope hereof. It should thus be noted that the matter containedin the above description or shown in the accompanying drawings should beinterpreted as illustrative and not in a limiting sense. The followingclaims are intended to cover all generic and specific features describedherein, as well as all statements of the scope of the present method andsystem, which, as a matter of language, might be said to falltherebetween.

What is claimed is:
 1. A time-multiplexed dual atomic magnetometer,comprising: first and second vapor cells located adjacent to a samplecell such that a magnetic field generated by a sample within the samplecell induces Larmor precession of atoms within the first and secondvapor cells; a first polarimeter that measures a first polarization of afirst probe beam after the first probe beam propagates through the firstvapor cell, the first polarimeter outputting a first polarization signalindicative of the first polarization; a second polarimeter that measuresa second polarization of a second probe beam after the second probe beampropagates through the second vapor cell, the second polarimeteroutputting a second polarization signal indicative of the secondpolarization; and a signal processor that processes alternating datablocks of the first and second polarization signals to generate a singlegapless temporal sequence that represents the magnetic field generatedby the sample.
 2. The time-multiplexed dual atomic magnetometer of claim1, wherein the signal processor: sequentially performsinstantaneous-phase retrieval on each of the alternating data blocks toobtain a corresponding magnetic-field sequence; and appends eachmagnetic-field sequence to the single gapless temporal sequence.
 3. Thetime-multiplexed dual atomic magnetometer of claim 2, wherein the signalprocessor outputs the single gapless temporal sequence.
 4. Thetime-multiplexed dual atomic magnetometer of claim 1, further comprisinga controller that gates the first and second probe beams such that: thefirst probe beam only propagates through the first vapor cell during afirst measurement stage to generate a first data block of thealternating data blocks; and the second probe beam only propagatesthrough the second vapor cell during a second measurement stage togenerate a second data block of the alternating data blocks, the secondmeasurement stage beginning when the first measurement stage ends. 5.The time-multiplexed dual atomic magnetometer of claim 4, wherein afirst duration of the first measurement stage is similar to a secondduration of the second measurement stage.
 6. The time-multiplexed dualatomic magnetometer of claim 4, wherein the controller gates a firstpump beam and a second pump beam such that: the first pump beam onlypropagates through the first vapor cell during a first pumping stageprior to the first measurement stage, the first pump beamspin-polarizing the atoms in the first vapor cell; and the second pumpbeam only propagates through the second vapor cell during a secondpumping stage prior to the second measurement stage, the second pumpbeam spin-polarizing the atoms in the second vapor cell.
 7. Thetime-multiplexed dual atomic magnetometer of claim 6, wherein a firstduration of the first pumping stage is similar to a second duration ofthe second pumping stage.
 8. The time-multiplexed dual atomicmagnetometer of claim 1, further comprising at least one magnetic fieldcoil that applies a magnetic bias field to the first vapor cell, thesecond vapor cell, and the sample cell.
 9. The time-multiplexed dualatomic magnetometer of claim 8, wherein: the first and second probebeams propagate along a common propagation direction; and the at leastone magnetic field coil is positioned such that the magnetic bias fieldis parallel to the common propagation direction.
 10. Thetime-multiplexed dual atomic magnetometer of claim 8, wherein: the firstand second probe beams propagate along a common propagation direction;and the at least one magnetic field coil is positioned such that themagnetic bias field is perpendicular to the common propagationdirection.
 11. A method for time-multiplexed dual atomic magnetometry,comprising: generating, by a sample within a sample cell, a magneticfield that induces Larmor precession of atoms within first and secondvapor cells located adjacent to the sample cell; measuring, with a firstpolarimeter, a first polarization of a first probe beam after the firstprobe beam propagates through the first vapor cell, the firstpolarimeter outputting a first polarization signal indicative of thefirst polarization; measuring, with a second polarimeter, a secondpolarization of a second probe beam after the second probe beampropagates through the second vapor cell, the second polarimeteroutputting a second polarization signal indicative of the secondpolarization; and processing alternating data blocks of the first andsecond polarization signals to generate a single gapless temporalsequence that represents the magnetic field generated by the sample. 12.The method of claim 11, wherein said processing includes: sequentiallyperforming instantaneous-phase retrieval on each of the alternating datablocks to obtain a corresponding magnetic-field sequence; and appendingeach magnetic-field sequence to the single gapless temporal sequence.13. The method of claim 12, further comprising outputting the singlegapless temporal sequence.
 14. The method of claim 11, furthercomprising gating the first and second probe beams such that: the firstprobe beam only propagates through the first vapor cell during a firstmeasurement stage to generate a first data block of the alternating datablocks; and the second probe beam only propagates through the secondvapor cell during a second measurement stage to generate a second datablock of the alternating data blocks, the second measurement stagebeginning when the first measurement stage ends.
 15. The method of claim14, wherein said gating the first and second probe beams includes gatingthe first and second probe beams such that a first duration of the firstmeasurement stage equals a second duration of the second measurementstage.
 16. The method of claim 14, further comprising gating a firstpump beam and a second pump beam such that: the first pump beam onlypropagates through the first vapor cell during a first pumping stageprior to the first measurement stage, the first pump beamspin-polarizing the atoms in the first vapor cell; and the second pumpbeam only propagates through the second vapor cell during a secondpumping stage prior to the second measurement stage, the second pumpbeam spin-polarizing the atoms in the second vapor cell.
 17. The methodof claim 16, wherein said gating the first and second pump beamsincludes gating the first and second pump beams such that a firstduration of the first pumping stage is similar to a second duration ofthe second pumping stage.
 18. The method of claim 11, further comprisingapplying a magnetic bias field to the first vapor cell, the second vaporcell, and the sample cell.
 19. The method of claim 18, wherein saidapplying includes applying the magnetic bias field such that a directionof the magnetic bias field is parallel to a common propagation directionof the first and second probe beams.
 20. The method of claim 18, whereinsaid applying includes applying the magnetic bias field such that adirection of the magnetic bias field is perpendicular to a commonpropagation direction of the first and second probe beams.